To mount/manipulate the DMG without conversion, you could use HFSExplorer. It comes as freeware, and has a decent set of options and features. Since your working on Windows 7, 7-Zip is the only other option I see for your situation. Either way, however, it appears as though you would have to perform some form of conversion since you're using Windows to access the dmg disk image archive. Then, you could use a preferred program of your choice to mount the new, workable disk image. I would also suggest avoiding TransMac unless you're willing to invest in a paid option. The closest I've come to doing what you're asking about involved the use of a third-party fs driver from Paragon solutions, but it only appears to work with universal disk images like the ISO.

In Windows, most dmg images can be opened using several other programs
such as Acute Systems TransMac, HFSExplorer, 7-Zip, UltraISO or
IsoBuster. MacDrive[9] can also mount simple dmg files as drives under
windows, but not sparse disk or encrypted dmgs.

I suggest you try 7-Zip as you can use it for free and it's pretty solid software.

For any Linux users wondering about this, the same Wikipedia page states:

In Linux and possibly other Unix flavors, most .dmg files can be
burned to CD/DVD using the program cdrecord or directly mounted to a
mountpoint (e.g. mount -o loop,ro -t hfsplus imagefile.dmg
/mnt/mountpoint).

PowerISO is a proprietary, cross platform, Freeware CLI tool that runs
on Linux; it can convert most proprietary image formats to ISO, and
newer versions support .dmg.